Forest polynomials and the class of the permutahedral variety

We study a basis of the polynomial ring that we call forest polynomials. This family of polynomials is indexed by a combinatorial structure called indexed forests and permits several definitions, one of which involves flagged P-partitions. As such, these polynomials have a positive expansion in the basis of slide polynomials. By a novel insertion procedure that may be viewed as a generalization of the Sylvester correspondence we establish that Schubert polynomials decompose positively in terms of forest polynomials. Our insertion procedure involves a correspondence on words which allows us to show that forest polynomials multiply positively. We proceed to show that forest polynomials are a particularly convenient basis in regard to studying the quotient of the polynomial ring modulo the ideal of positive degree quasisymmetric polynomials. This aspect allows us to give a manifestly nonnegative integral description for the Schubert class expansion of the cohomology class of the permutahedral variety in terms of a parking procedure. We study the associated combinatorics in depth and introduce a multivariate extension of mixed Eulerian numbers.